A hyperbolic spiral is a

plane curve In mathematics, a plane curve is a curve in a plane that may be either a Euclidean plane, an affine plane or a projective plane. The most frequently studied cases are smooth plane curves (including piecewise smooth plane curves), and algebraic pla ...

, which can be described in polar coordinates by the equation :

r=\frac

of a

hyperbola In mathematics, a hyperbola (; pl. hyperbolas or hyperbolae ; adj. hyperbolic ) is a type of smooth curve lying in a plane, defined by its geometric properties or by equations for which it is the solution set. A hyperbola has two pieces, cal ...

. Because it can be generated by a circle inversion of an

Archimedean spiral The Archimedean spiral (also known as the arithmetic spiral) is a spiral named after the 3rd-century BC Greek mathematician Archimedes. It is the locus corresponding to the locations over time of a point moving away from a fixed point with a con ...

, it is called Reciprocal spiral, too..

Pierre Varignon Pierre Varignon (1654 – 23 December 1722) was a French mathematician. He was educated at the Jesuit College and the University of Caen, where he received his M.A. in 1682. He took Holy Orders the following year. Varignon gained his first ex ...

first studied the curve in 1704. Later

Johann Bernoulli Johann Bernoulli (also known as Jean or John; – 1 January 1748) was a Swiss mathematician and was one of the many prominent mathematicians in the Bernoulli family. He is known for his contributions to infinitesimal calculus and educating L ...

and

Roger Cotes Roger Cotes (10 July 1682 – 5 June 1716) was an English mathematician, known for working closely with Isaac Newton by proofreading the second edition of his famous book, the '' Principia'', before publication. He also invented the quadratur ...

worked on the curve as well. The hyperbolic spiral has a pitch angle that increases with distance from its center, unlike the logarithmic spiral (in which the angle is constant) or Archimedean spiral (in which it decreases with distance). For this reason, it has been used to model the shapes of

spiral galaxies Spiral galaxies form a class of galaxy originally described by Edwin Hubble in his 1936 work ''The Realm of the Nebulae'' From vector calculus in polar coordinates one gets the formula for the ''polar slope'' and its angle between the tangent of a curve and the tangent of the corresponding polar circle. For the hyperbolic spiral the ''polar slope'' is :

\tan\alpha=-\frac.

Curvature

The curvature of a curve with polar equation is :

\kappa = \frac .

From the equation and the derivatives and one gets the ''curvature'' of a hyperbolic spiral: :

\kappa(\varphi) = \frac.

Arc length

The length of the arc of a hyperbolic spiral between and can be calculated by the integral: :

^ . \end

Sector area

The area of a sector (see diagram above) of a hyperbolic spiral with equation is: :

\begin
A&=\frac12\int_^ r(\varphi)^2\, d\varphi\\
&=\frac12\int_^\frac\, d\varphi\\
&= \frac\left(\frac-\frac\right)\\
&=\frac\bigl(r(\varphi_1)-r(\varphi_2)\bigr) .
\end

Inversion

The inversion at the unit circle has in polar coordinates the simple description: . The image of an Archimedean spiral with a circle inversion is the hyperbolic spiral with equation . At the two curves intersect at a fixed point on the unit circle. The

osculating circle In differential geometry of curves, the osculating circle of a sufficiently smooth plane curve at a given point ''p'' on the curve has been traditionally defined as the circle passing through ''p'' and a pair of additional points on the curve i ...

of the Archimedean spiral at the origin has radius (see

) and center . The image of this circle is the line (see

circle inversion A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre. Equivalently, it is the curve traced out by a point that moves in a plane so that its distance from a given point is const ...

). Hence the preimage of the asymptote of the hyperbolic spiral with the inversion of the Archimedean spiral is the osculating circle of the Archimedean spiral at the origin. :''Example:'' The diagram shows an example with .

Central projection of a helix

Consider the central projection from point onto the image plane . This will map a point to the point . The image under this projection of the helix with parametric representation :

(r\cos t, r\sin t, ct),\quad c\neq 0,

is the curve :

\frac(\cos t,\sin t)

with the polar equation :

\rho=\frac,

which describes a hyperbolic spiral. For parameter the hyperbolic spiral has a pole and the helix intersects the plane at a point . One can check by calculation that the image of the helix as it approaches is the asymptote of the hyperbolic spiral.

References

* Hans-Jochen Bartsch, Michael Sachs: ''Taschenbuch mathematischer Formeln für Ingenieure und Naturwissenschaftler'', Carl Hanser Verlag, 2018, , 9783446457072, S. 410. * Kinko Tsuji, Stefan C. Müller: ''Spirals and Vortices: In Culture, Nature, and Science'', Springer, 2019, , 9783030057985, S. 96. * Pierre Varignon
''Nouvelle formation de Spirales – exemple II'', Mémoires de l’Académie des sciences de l’Institut de France, 1704, pp. 94–103.
*

Friedrich Grelle Friedrich may refer to: Names *Friedrich (surname), people with the surname ''Friedrich'' *Friedrich (given name), people with the given name ''Friedrich'' Other *Friedrich (board game), a board game about Frederick the Great and the Seven Years' ...

: ''Analytische Geometrie der Ebene'', Verlag F. Brecke, 186
hyperbolische Spirale
S. 215. *

Jakob Philipp Kulik Jakob Philipp Kulik (1793–1863) was an Austrian mathematician known for his construction of a massive factor tables. Biography Kulik was born in Lemberg, which was part of the Austrian empire, and is now Lviv located in Ukraine. Kulik's fa ...

: ''Lehrbuch der höhern Analysis, Band 2'', In Commiss. bei Kronberger u. Rziwnatz, 1844
Spirallinien
S. 222.

External links

*
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{{Spirals Spirals pt:Espiral logarítmica